Question

hyperbolas
you are looking for the foci of the graph $\frac{x^2}{4} - \frac{y^2}{9} = 1$. so far, you have determined that $a^2 = 4$, $b^2 = 9$, and $c = sqrt{13}$.
what are the foci of the graph $\frac{x^2}{4} - \frac{y^2}{9} = 1$?
a. $(0, pm sqrt{13})$
b. $(pm 4, 0)$
c. $(pm sqrt{13}, 0)$
d. $(0, pm 9)$

Explanation:

Step1: Identify hyperbola orientation

The given equation $\frac{x^2}{4}-\frac{y^2}{9}=1$ follows the horizontal transverse axis hyperbola form $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, so foci lie on the x-axis.

Step2: Confirm focus coordinates formula

For this hyperbola, foci are at $(\pm c, 0)$, where $c=\sqrt{a^2+b^2}$.

Step3: Substitute known values

We know $a^2=4$, $b^2=9$, so $c=\sqrt{4+9}=\sqrt{13}$. Thus, foci are $(\pm\sqrt{13},0)$.

Answer:

C. $(\pm\sqrt{13},0)$